Approaches to poker tournament winner prediction?
This looks like it maps very nicely onto Association Mining. In association mining, you are trying to find items from a discrete set that often co-occur in transactions. For instance, you might want to find the items that most commonly appear in an online shopping cart together.

In your case, the problem amounts to:


Split up the data into sub-problems by who won.
Perform association mining on the sets of players in each sub-problem, using, e.g. the apriori algorithm. 


The resulting rules will then have associated confidence factors. When you want to predict who will win, you can take the winner suggested by the rule with the greatest confidence.



The other approach suggested by your problem is to model it as classification. Here you are trying to assign labels (who won) to input vectors (who played). The process would look like:


Map the sets of who played into binary features. So the subset P1,P2, P4,P8 might be represented with {1,1,0,1,0,0,0,1} if there are only 8 players. Map the winning player onto a numeric class (e.g. the numbers 1-8). 
Run any classification algorithm to create a model that predicts the class from the binary features. A decision tree learner might be a interesting starting point if you want to understand which factors are important to the model. There are many other techniques however.


You can use the model you train to make predictions about future games with the same set of players.